Questions tagged [cauchy-product]

For questions about the Cauchy product, the discrete convolution of two sequences.

Filter by
Sorted by
Tagged with
1
vote
3answers
53 views

Is there a way to re-write $\sum_{n=1}^{\infty}{\left(\sum_{k=1}^{n}{\frac{1}{k}}\right)}z^n$

Is there a way to re-write $\sum_{n=1}^{\infty}{\left(\sum_{k=1}^{n}{\frac{1}{k}}\right)}z^n$ I was thinking that I could use the cauchy product, but therefore I have to achieve this:$$\left(\sum_{...
0
votes
2answers
31 views

How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$

How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$ I believe that I have to use the cauchy product? But how do transform the expression to be a product ...
0
votes
1answer
24 views

Product of Two Summations with Different Upper Limits

I'm trying to multiply two different finite summations with different upper limits. I've tried Cauchy Product but i think it's valid for same upper limits. I also tried to split the summation. Any ...
0
votes
3answers
50 views

Find the power series of $f(x)=\frac{x}{x+1}$ in $x_0=0$

In 1st Semester Calculus book I found an exercise that asks me to find the above power series of the function at the point $x_0 = 0$ using the geometric series formula and the Cauchy-Product. So far I'...
0
votes
1answer
49 views

Cauchy product and geometric series [duplicate]

I was given this series: Let $q \in \mathbb{C}, \mid q\mid <1. $ $$\frac{1}{2}\sum_{n=0}^{\infty} (n^2 +3n +2)q^n $$ Now I have to show that $$\frac{1}{(1-q)^3}=\frac{1}{2}\sum_{n=0}^{\infty} (...
0
votes
1answer
41 views

Multiplying two polynomials: explanation of the general formula for the coefficients

If $$f(x)=a_nx^n+...+a_0$$ and $$g(x)=b_mx^m+…+b_0$$ then $$f(x)\cdot g(x)=c_{m+n}x^{m+n}+...+c_0$$ where $c_k=\sum_{r+s=k}a_rb_s,\quad k=0,....,m+n$ I know that the degree $0$ and $(m+n)$ exists, ...
2
votes
2answers
43 views

Generating Function For Catalan Numbers Type Sequence

I've been working my way through an old post, but I don't think the solution offered can be correct. The question is; Find the generating function (within a choice of sign) for: $$c_{n+1} = 2\sum_{k=...
1
vote
0answers
35 views

Cauchy product of power series with $x^{2n}$

I am trying to rewrite a function $y(x)=\frac{1}{1+x+x^2+x^3}$ as a series. I used geometric series and got to two power series that I need to Cauchy product, however, one of them has $x^{2n}$ and I ...
1
vote
0answers
62 views

Infinitely Nested Infinite Series / Infinite Composition of Series

Is there any documentation about a series like this? What is it called? Does it have a value? I have tried several searches and couldn't find anything close to this. Any guidance would be ...
0
votes
0answers
34 views

Cauchy product for the reciprocal of the polynomial $x - 2x^2 + 3x^3 - 4x^4$

I have come across some Laurent series in which the denominator of a fraction contains a power series. Looking around I came across this Calculate Laurent series for $1/ \sin(z)$ which suggests that ...
1
vote
1answer
23 views

How was Merten's theorem proven on the step comparing the convergence of one sequence and the Cauchy product partial sum?

I'm trying to understand step 3 of the proof of Merten's theorem. How is it known that $N*_{sup}|B_i-B|$ term converges slow enough that there is an N where equation 3 is true: $|a_n|\le\frac{\...
1
vote
2answers
70 views

Proving $\sin(2x) = 2\sin(x)\cos(x)$ using Cauchy product

I've stumbled upon this question. I can't understand why all even terms of the Cauchy product are $0$, since we add only positive numbers: \begin{equation*} c_n = \begin{cases} \sum\limits_{k=0}^{m} \...
5
votes
1answer
136 views

On the series expansion of $\frac{\operatorname{Li}_3(-x)}{1+x}$ and its usage

Recently I have asked about the evaluation of an integral involving a Trilogarithm $($which can be found here$)$. Pisco provided a quite elegant approach starting with a functional equation of the ...
4
votes
2answers
77 views

Simplifying $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{l=0}^{\min(n,m)}a_{l}b_{m-l}c_{n-l}$

I have come across a sum of the following form; $$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{l=0}^{\min(n,m)}a_{l}b_{m-l}c_{n-l}$$ and want to simplify it (in particular to remove the $min(n,m)$). ...
6
votes
0answers
316 views

Series whose Cauchy product is absolutely convergent - A general example

Is there series that is divergent or conditionally convergent with absolutely convergent Cauchy product? Seems like there is a group of these examples! Perhaps finding divergent series with ...
2
votes
1answer
126 views

Conditionally converges $\sum_{k=0}^\infty a_n$, $\sum_{k=0}^\infty b_n$ and also their Cauchy product

Edited: I post a new post here that is somewhat related to this question. It proved that: If $\sum_{k=0}^\infty a_n$ and $\sum_{k=0}^\infty b_n$ converges conditionally (not absolutely), then their ...
0
votes
1answer
33 views

Calculating the limit based on the cauchy product

I'm attempting to find the limit of $$ \sum^\infty_{k=0} k^2 q^k$$ using the result of the cauchy product $$ \sum^\infty_{k=0} k q^k * \sum^\infty_{k=0} q^k$$ and I have calculated the cauchy ...
1
vote
3answers
77 views

Summation Recurrence Relation

How to solve this Summation Recurrence Relation: $$x_n=\sum_{i=1}^n a_ix_{n-i}\,,\,\,\,n\ge1$$where, $x_0=1$ and $a_n$ is some arbitrary sequence. The right hand side of the recurrence looks ...
1
vote
1answer
75 views

Radius of convergence of Cauchy product – examples

Assume two power series $\sum_{n\ge0}a_n x^n=f_a(x),\sum_{n\ge0}b_n x^n=f_b(x)$ with radii $r_a,r_b$ (respectively) and $r_a\lneqq r_b.$ Consider their Cauchy product $$\sum_{n\ge0}\left(\sum_{k=0}^n ...
0
votes
2answers
93 views

Find the Cauchy product of $e^x$ and $e^{-x}$.

I was able to get to: $c_n$ = $$\sum_{k=0}^{n} \binom{n}{k} \frac{x^k (-x)^{n-k}}{n!}$$ but I am stuck as to how to get past this. Is there a property of the series that lets me pull the negative ...
5
votes
2answers
156 views

How to get sums like these in the form of the Cauchy Product?

I know that the question isn't very well-worded, please feel free to change it to something better. I have this sum: $$\sum_{k=0}^{\infty} \sum_{l=0}^{k} a_lx^ll!a_{k-l}x^{k-l+1}(k-l)!\frac{1}{(k+1)!...
2
votes
1answer
113 views

Counterexample to Cauchy product theorem

The Cauchy product theorem for infinite series of complex numbers states if $\sum a_n$ and $\sum b_n$ are two absolutely convergent series then the Cauchy product $\sum c_n$, where $c_n=\sum_{p+q=n} ...
0
votes
0answers
59 views

Cauchy Product of Two Divergent Series

Problem Examine the following Cauchy product and the factors for convergence and in the case of convergence determine the limit. $$\left(3 + \sum^{\infty}_{k=1}3^k\right)\left(-2+\sum^{\infty}_{k=1}...
0
votes
0answers
13 views

Unclear equation in my scriptum (perhaps cauchy-product) [duplicate]

Hi the following equation in my scriptum seems unclear. I think it has something to do with cauchy-product but i dont know
1
vote
1answer
90 views

How to recast these two exponential of infinite power series as simple power series?

I have two exponential of infinite power series, with different expressions for coefficients $a_n$, that I would like to recast as two other power series without the exponential $$\exp\left(\sum_{n=1}^...
1
vote
4answers
49 views

Power series of $\frac{1}{(1-z)^2}$

I want to show, that the following is true for every $z\in C$ with $|z|<1$: $$\frac{1}{(1-z)^2} =\sum_{k=1}^\infty kz^{k-1}$$ I think there is a way with the Cauchy-Product
1
vote
1answer
50 views

Let $G(x)=\frac{1}{(1-x)^2}$. Prove that $G(x)=\sum_{n=0}^{\infty}(n+1)x^n$.

Let $G(x)=\frac{1}{(1-x)^2}$. Prove that $$G(x)=\sum_{n=0}^{\infty}(n+1)x^n.$$ The solution given uses the Cauchy Product. It is shown below: $$\begin{aligned} G(x)&=\left(\sum_{n=0}^{\infty}x^n\...
1
vote
0answers
442 views

Power Series Solution for an ODE which has trigonometric coefficient functions

The ODE for which we seek a power series solution is: $$y''+ \cos(x)y' + x\sin(x)y = 0,\hspace{0.4cm} y(0) = 1,\hspace{.1cm} y'(0) = 0$$ I need to find the partial sum up to five, from the initial ...
3
votes
3answers
197 views

Cauchy Product starting from $1$

The definition of the Cauchy product from Wikipedia is defined as $$\left(\sum_{i=0}^\infty a_i\right)\left(\sum_{j=0}^\infty b_j\right) =\sum_{k=0}^\infty\sum_{\ell=0}^ka_\ell b_{k-\ell}$$ ...
3
votes
2answers
55 views

show that $\left(\sum_{i=0}^\infty\frac{a^i}{i!}\right)\left(\sum_{j=0}^\infty\frac{b^j}{j!}\right) = \sum_{k=0}^\infty\frac{(a+b)^k}{k!}$

I need to show that if $\sum_{i=0}^\infty \frac{a^i}{i!}$ is absolutely convergent for all $a\in\mathbb{R}$, then $$\left(\sum_{i=0}^\infty\frac{a^i}{i!}\right)\left(\sum_{j=0}^\infty\frac{b^j}{j!}\...
1
vote
0answers
61 views

Decomposition in product

I start my question with an example that I did. Give the following series $R(x):=\sum_{d=1}^{\infty}(\sum_{l=0}^{d-1} (-1)^{l}\frac{1}{ \ \ell!(d-1-\ell)!})q_{1}^{d} x^d$. I could decompose this as a ...
3
votes
2answers
145 views

Calculate sum of a series

How to find the sum of $$\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}\left\{\sum_{k=1}^{2n+1}\frac{(-1)^k} k \right\}$$ $$\begin{array}\\ \frac{1}{1}&\times&(-\frac{1}{1}) &+&\ \ \\ (-\...
5
votes
4answers
173 views

Find limit of sum

I suspect that $\lim_{n \to \infty} \sum_{k = 0}^{n - 1}\frac{k}{a^k(n - k)} = 0$ for $a > 1$. I know that this product represents the taylor coefficients of $\frac{-ax\ln(1 - x)}{(a - x)^2}$ by ...
3
votes
3answers
110 views

The series $\sum_{k=1}^\infty\sum_{n=1}^k k^{-3}/(2n-1)$

$$\sum_{k=1}^\infty\sum_{n=1}^k\frac{1}{(2n-1)k^3}$$ Can anyone help me find this series? I tried to use Cauchy product but I don't know how I can complete it.
4
votes
2answers
404 views

Convergent Cauchy product of divergent series

I was looking at the counterexample $a_n = b_n = (-1)^n/\sqrt{n}$ where $\sum a_n$ and $\sum b_n$ converge but the Cauchy product $\sum_{k=0}^\infty c_k = \sum_{k=0}^\infty \sum_{j=0}^{k}a_j b_{k-j}$ ...
3
votes
2answers
254 views

Suppose $\sum a_n$ converges absolutely and $\sum b_n$ converges. Give an example where the Cauchy product does not converge absolutely.

Problem: Suppose $\sum_{n=0}^{\infty} a_n$ converges absolutely and $\sum_{n=0}^{\infty} b_n$ converges. Give an example where the Cauchy product, defined as $\sum_{n=0}^{\infty} c_n$ where $c_n=\...
0
votes
0answers
188 views

Cauchy product of series, all three series convergent

Let $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ two (not necessarily absolute) convergent series. We denote their Cauchy product by $\sum_{n=0}^\infty c_n$ where $c_n=\sum_{l=0}^n a_lb_{k-l}$...
0
votes
0answers
635 views

Product of absolutely convergent series is absolutely convergent

I know there are a couple of proofs of this result on this website, but I would like to verify if my proof is correct, as it is slightly different. Let $\sum\limits_{n=0}^\infty a_n$ and $\sum\...
3
votes
4answers
346 views

Cauchy product of $\sum\limits_n^{\infty}\frac{1}{n}$ with itself [closed]

Is the cauchy product of $\sum\limits_n^{\infty}\frac{1}{n}$ with itself simply $\sum\limits_n^{\infty}\frac{1}{n^2}$? I can't apply the definition here https://proofwiki.org/wiki/Definition:...
0
votes
0answers
143 views

Double series, Cauchy product

I'm totally stuck: $a_n \in \mathbb{C} $ is defined for $n\in \mathbb{Z} $ and $\{ b_k\}_{k \in \mathbb{N}}$ is a sequence. The infinite sums $ \sum_{n\in \mathbb{Z}} a_n := \lim_{N \to \infty} \...
0
votes
1answer
433 views

Example of Cauchy product of two series with radius of convergence $\rho'> \min\{\rho_1,\rho_2\}$

Can anyone suggest a (relatively simple) example of two real power series $\sum_{n \geq 0} a_n x^n$ and $\sum_{n \geq 0} b_n x^n$, with radii of convergence rispectively $\rho_1$ and $\rho_2$, whose ...
2
votes
0answers
50 views

Estimating cauchy products of divergent series

Let $\sum a_n,\sum b_n\in\mathbb{R}_+$ be two series and let $F(x),G(x)$ be their corresponding generating functions, that is, $$ F(x)=\sum_{n=0}^\infty a_nx^n\\ G(x)=\sum_{n=0}^\infty b_nx^n$$ In ...
1
vote
1answer
703 views

Sum of exponential series

Are there any ways to transform the product in $$y=\left(\sum_ {k=1}^N a_k \exp\left(it \mu_k-\frac{\sigma^2_k t^2}{2}\right)\right) \times \left(\sum_ {k=1}^M b_k \exp\left(it \nu_k-\frac{\Sigma^...
4
votes
2answers
167 views

On the asymptotic behaviour of the Cauchy product of harmonic numbers $\sum_{k=1}^n H_k H_{n-k+1}$

In this post we take in our hands two simple tools. The first is the generating function of the harmonic numbers, you can see it in this Wikipedia, section 3, that holds for $|z|<1$. The second is ...
0
votes
1answer
57 views

Sum of Square using Cauchy Product

Some background on the question: I am trying to compute the expectation of $\mathbb E[(Y|X)^2]$ of Y conditional on X which follows an exponential distribution ~(λ/k) So the question boils down to ...
3
votes
1answer
479 views

Proving $\sin 2x = 2\sin x \cos x$ using Taylor Series and Cauchy products

I have that the Taylor series of $\sin x$ and $\cos x$ are \begin{equation*} \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\ \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}x^{2n} \end{...
1
vote
0answers
165 views

Bernoulli numbers, primes, integers and the Möbius function

For $|\frac{x}{2\pi}|<1$ using Möbius inversion for the Taylor series of the logarithm one deduces $$\frac{x}{2\pi}=-\sum_{n=1}^\infty\frac{\mu(n)}{n}\log\left(1-\frac{x^n}{(2\pi)^n}\right)$$ thus $...
1
vote
0answers
52 views

Imposing Condition on a Cauchy Product

Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is an integrable function with Fourier coefficients given by $\hat{f}$. Then, since $|f|^2 = f \cdot \bar{f}$, we have: $$\displaystyle \int_{0}^{2\...
2
votes
0answers
218 views

Product of a Multinomial done using Multinomial Theorem?

I was looking into the multinomial theorem to try to understand more about summing a sequence over a composition of a number with non-negative elements (a weak composition). In particular, I am ...
0
votes
2answers
138 views

Prove or disprove: The Cauchy-product of serieses $\sum_{k=0}^{\infty}\frac{1}{k!}$ and $\sum_{k=0}^{\infty}\frac{2^{k}}{k!}$ converges to $e^{2}$

Prove or disprove: The Cauchy-product of serieses $\sum_{k=0}^{\infty}\frac{1}{k!}$ and $\sum_{k=0}^{\infty}\frac{2^{k}}{k!}$ converges to $e^{2}$. I'm not really sure if it's done like that but I ...